Link diagrams

Introduction

Link diagrams are a fundamental concept in the field of knot theory, a branch of topology that studies mathematical knots. Unlike physical knots, which are tangible and can be tied and untied, mathematical knots are abstract objects that exist in three-dimensional space. A link diagram is a two-dimensional representation of a link, which is a collection of one or more knots that may be interlinked or separate. These diagrams are crucial for visualizing and analyzing the properties of links, as they provide a way to study the spatial arrangement and interactions between the components of a link.

Historical Background

The study of link diagrams can be traced back to the late 19th century, when pioneering mathematicians such as Peter Guthrie Tait and James Clerk Maxwell began exploring the mathematical properties of knots and links. Tait, in particular, made significant contributions to the development of knot theory by cataloging knots and links based on their diagrams. His work laid the foundation for modern knot theory and inspired future generations of mathematicians to delve deeper into the study of links and their diagrams.

Construction of Link Diagrams

Link diagrams are constructed by projecting a link onto a plane, resulting in a two-dimensional representation that captures the essential features of the link. The projection process involves choosing a direction from which to view the link, and then drawing the resulting image on a plane. The key elements of a link diagram are the crossings, where one strand of the link passes over or under another strand. These crossings are typically represented by breaks in the strands, with the overstrand drawn as an unbroken line and the understrand as a broken line.

Reidemeister Moves

A crucial aspect of link diagrams is their invariance under Reidemeister moves, which are a set of three local transformations that can be applied to a link diagram without changing the topological type of the link. These moves, named after the mathematician Kurt Reidemeister, are essential for understanding the equivalence of different link diagrams. The three Reidemeister moves are:

1. **Twist Move (Type I):** Adding or removing a twist in a single strand. 2. **Poke Move (Type II):** Sliding one strand over another, creating or removing a pair of crossings. 3. **Slide Move (Type III):** Sliding a strand over a crossing between two other strands.

These moves allow mathematicians to manipulate link diagrams and demonstrate the equivalence of different representations of the same link.

Types of Links and Their Diagrams

Links can be classified into various types based on their structure and properties. Some common types of links include:

Unlinks

An unlink is a collection of disjoint, non-intersecting loops. In a link diagram, an unlink is represented by a series of non-overlapping circles. The simplest example of an unlink is the trivial link, which consists of a single unknotted loop.

Hopf Links

A Hopf link is the simplest nontrivial link, consisting of two interlinked loops. In a link diagram, the Hopf link is represented by two circles that intersect at two crossings. The Hopf link is significant because it is the simplest example of a link that cannot be separated into individual components without cutting the strands.

Borromean Rings

The Borromean rings are a set of three interlinked loops that are arranged in such a way that removing any one loop results in the other two loops becoming unlinked. In a link diagram, the Borromean rings are represented by three circles that intersect at multiple crossings. This configuration is notable for its nontrivial linking properties, as it demonstrates that the whole is greater than the sum of its parts.

Torus Links

Torus links are a family of links that can be embedded on the surface of a torus. These links are characterized by two parameters, p and q, which determine the number of times the link winds around the torus in each direction. In a link diagram, torus links are represented by a series of interlocking loops that follow a regular pattern.

Link Invariants

Link invariants are mathematical quantities that remain unchanged under Reidemeister moves, making them useful tools for distinguishing between different links. Some important link invariants include:

Linking Number

The linking number is an integer that measures the number of times one component of a link winds around another component. It is calculated by counting the signed crossings between the components in a link diagram. The linking number is an important invariant for distinguishing between different types of links, as it provides information about the degree of entanglement between the components.

Jones Polynomial

The Jones polynomial is a polynomial invariant of links that assigns a polynomial to each link diagram. It is calculated using a recursive process that involves applying specific rules to the crossings in the diagram. The Jones polynomial is a powerful tool for distinguishing between different links, as it captures information about the topology of the link that is not apparent from the diagram alone.

Alexander Polynomial

The Alexander polynomial is another polynomial invariant of links, which is derived from the fundamental group of the link complement. It is calculated using a presentation of the link group and provides information about the topology of the link. The Alexander polynomial is particularly useful for studying the properties of knots and links, as it can be used to detect certain types of symmetries and other topological features.

Applications of Link Diagrams

Link diagrams have a wide range of applications in both mathematics and other scientific fields. Some notable applications include:

DNA Topology

In the field of molecular biology, link diagrams are used to study the topology of DNA molecules. DNA can form complex structures, such as knots and links, which can affect its biological function. By representing DNA as a link diagram, researchers can analyze the topological properties of DNA and gain insights into its behavior and interactions.

Quantum Computing

Link diagrams also play a role in the field of quantum computing, where they are used to study the properties of quantum entanglement. Quantum entanglement is a phenomenon in which the quantum states of two or more particles become correlated, and link diagrams provide a way to visualize and analyze these correlations. This has led to the development of new quantum algorithms and protocols that leverage the topological properties of links.

Chemistry

In chemistry, link diagrams are used to study the topology of molecular structures. Certain molecules, known as catenanes and rotaxanes, have topologies that resemble links and knots. By representing these molecules as link diagrams, chemists can analyze their properties and interactions, leading to the development of new materials and chemical processes.